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Chapter 8: Problem 3

Consider a rigid molecule made of \(N\) segments. Let \(\zeta\) be the friction constant of the molecule, i.e., the frictional force exerted on the molecule when it moves with velocity \(v\) in a solvent is given by \(-\zeta v\). Consider a dilute solution of such molecules. The friction constant \(\xi\) per unit volume of such solution is given by \(\xi=\frac{\pi}{N} \zeta\). Assuming that the approximation of eq. (8.39) is valid, answer the following questions. (a) Show that the friction constant \(\zeta\) per molecule is given by $$ \frac{1}{\zeta}=\frac{1}{6 \pi \eta N} \int d r \frac{g(r)}{|r|} $$ (b) Let \(r_{n},(n=1,2, \ldots, N)\) be the position vector of the segments. Show that \(\zeta\) can be written as $$ \frac{1}{\zeta}=\frac{1}{6 \pi \eta N^{2}} \sum_{n} \sum_{n \neq m} \frac{1}{\left|r_{n}-r_{m}\right|} $$ (c) Calculate the friction constant \(\zeta\) of a sphere of radius \(a\) using eq. \((8.131)\) and show that it is given by \(\zeta=6 \pi \eta a\). (Hint: assume that the sphere is made of friction points which are uniformly distributed on the surface, then $$ \left.\frac{1}{\zeta}=\frac{1}{6 \pi \eta\left(4 \pi a^{2}\right)^{2}} \int d S \int d S^{\prime} \frac{1}{\left|r-r^{\prime}\right|}\right) $$ (d) Calculate the friction constant \(\zeta\) of a rod-like particle of length \(L\) and diameter \(b\) assuming that the rod is made of \(N=L / b\) friction points. (Answer: the result is $$ \left.\zeta=\frac{3 \pi \eta L}{\ln (L / b)} \quad \text { for } \quad L \gg b\right) $$

Short Answer

Expert verified
Show relationship between position vectors and overall friction constant for segments. Calculate the friction constant of a sphere and a rod using provided integrals.

Step by step solution

01

Understand the given terms and equations

Define the friction constant per molecule \(\zeta\) and the friction constant per unit volume \(\xi\). Understand the basic relationships given, such as \(\xi = \frac{\pi}{N} \zeta\).
02

Show the friction constant per molecule

To show \( \frac{1}{\zeta} = \frac{1}{6 \pi \eta N} \int d r \frac{g(r)}{|r|} \), consider the integration over the pair correlation function \(g(r)\), which accounts for the distribution of segments.
03

Relate the position vectors of segments

Given \ r_n \(n=1,2,...,N\), show using summation: \( \frac{1}{\zeta} = \frac{1}{6 \pi \eta N^2} \sum_{n} \sum_{n eq m} \frac{1}{|r_n - r_m|} \). This involves showing how the interaction between all pairs of segments contributes to the total friction constant.
04

Show friction constant of a sphere

Assume the sphere is composed of uniformly distributed friction points and use the given hint to show: \( \zeta = 6 \pi \eta a \). This involves evaluating a double surface integral over the spherical surface.
05

Calculate friction constant of a rod-like particle

For a rod-like molecule of length \(L\) and diameter \(b\), with \(N = L / b\) segments, use the given equation: \zeta = \frac{3 \pi \eta L}{\ln (L / b)}\ for \( L \gg b \). Simplify this using the properties of logarithms for large aspect ratios.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Friction Constant Per Molecule
When a molecule moves through a solvent, it experiences a resistance force due to the interactions with the solvent molecules. This force, known as frictional force, is captured by the friction constant \(\zeta\). The concept of the friction constant per molecule is essential to understand how individual segments of a molecule contribute to the overall friction experienced by the molecule. Specifically, the friction constant per molecule is given by \(\frac{1}{\zeta} = \frac{1}{6 \pi \eta N} \int d r \frac{g(r)}{\|r\|}\). Here, \(\eta\) represents the viscosity of the solvent and \(g(r)\) is the pair correlation function which describes how pairs of segments are distributed relative to each other.
This relationship highlights the interplay between the molecular segments and their spatial distribution.
Pair Correlation Function
The pair correlation function, denoted as \(g(r)\), is a crucial concept in understanding molecular friction in solvents. It essentially describes how the presence of one particle at a certain location affects the probability of finding another particle at a different location relative to it. This function is integral to determining the friction constant because it encapsulates the spatial distribution of interacting segments within the molecule. For example, if segments of a molecule are closer together, the function \(g(r)\) alters, significantly impacting \(\zeta\). Integrating \(g(r)\) over all possible inter-segment distances gives insight into how these spatial relationships contribute to the total friction experienced by the molecule.
Spherical Friction Model
To understand friction on spherical bodies in solvents, we need to consider the spherical friction model. For a sphere of radius \(a\), moving through a solvent, the friction constant \(\zeta\) is derived using the equation \(\zeta = 6 \pi \eta a\). This model assumes that the sphere is made up of infinitesimal friction points that are uniformly distributed over its surface. This distribution allows for a simplified computation of the friction coefficient by integrating these point interactions over the entire spherical surface. The end result is that the resistance force experienced by a spherical particle is directly proportional to its radius and the solvent's viscosity, capturing the essence of Stokes' law in fluid dynamics.
Rod-like Particle Friction
Rod-like particles exhibit different frictional properties compared to spherical particles due to their elongated shape. For a rod-like particle with length \(L\) and diameter \(b\), the friction constant \(\zeta\) is given by \(\zeta = \frac{3 \pi \eta L}{\ln(L / b)}\) for \(L \gg b\). In this context, the length-to-diameter ratio plays a significant role. As the length increases relative to the diameter, the frictional force grows logarithmically. This model is particularly useful in describing the behavior of long, thin particles moving through a fluid, where the friction depends predominantly on the length of the particle but is moderated by its slenderness.
Velocity-Dependent Friction Force
The velocity-dependent friction force is foundational to understanding dynamic molecular behavior in solvents. As a molecule moves with velocity \(v\), it experiences a frictional force that opposes its motion, given by \-\zeta v\. Here, \(\zeta\) is the friction constant derived from properties of the molecule and the solvent. This force ensures that higher velocities result in proportionally greater friction, thereby maintaining a balance between the driving force and solvent resistance. This linear relationship between frictional force and velocity is crucial for phenomena such as diffusion and sedimentation, where the resistance encountered by moving particles needs to be precisely quantified to predict their behavior accurately.

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Most popular questions from this chapter

Consider a single polymer chain in a dilute solution. The scattering function of the polymer chain is given by $$ S(k)=\frac{1}{N} \sum_{n \cdot m}\left\langle e^{i k \cdot\left(r_{n}-r_{m}\right)}\right\rangle $$ where \(n\) and \(m\) denote the segments constituting the polymer chain and \(N\) is the number of segments in the chain. Answer the following questions. (a) For small \(k\), the right-hand side of eq. (8.124) can be expanded as a power series of \(k\). Show that in the isotropic state, \(S(k)\) is written as $$ S(k)=N\left[1-\frac{1}{3} k^{2} R_{9}^{2}+\cdots\right] $$ where \(R_{g}^{2}\) is defined by $$ R_{g}^{2}=\frac{1}{2 N^{2}} \sum_{n, m}\left\langle\left(r_{n}-r_{m}\right)^{2}\right\rangle $$ \(R_{g}\) is called the radius of gyration of the polymer. \(\left(R_{g}\right.\) is the same as the correlation length \(\ell_{c}\) defined by eq. \((8.20)\). In polymer science, it is customary to use \(R_{g}\) rather than \(\ell_{c}\).) (b) Calculate the radius of gyration \(R_{g}\) of a linear Gaussian polymer. (Hint: use the fact that \(\left.\left\langle\left(r_{n}-r_{m}\right)^{2}\right\rangle=|n-m| b^{2} .\right)\) (c) Calculate \(R_{g}\) for a rigid rod-like polymer in which the segments are uniformly distributed along a line of length \(L\). Also calculate \(R_{g}\) for a globular polymer in which the segments are uniformly distributed in a sphere of radius \(R\). (d) Show that the fractal dimension of a flexible polymer is 2 , while that of rod-like polymer is 1 .

Consider a two-component solution near the critical point. Let \(\phi_{c}\) and \(T_{c}\) be the concentration and the temperature at the critical point. The free energy of the system is given by eq. (8.49) (with \(\rho_{1}=0\) ). If the concentration \(\phi\) is not far from \(\phi_{c}\), the free energy density \(f(\phi)\) can be written as \(f(\phi)=c_{0}(T)+c_{1}(T)\left(\phi-\phi_{c}\right)+\frac{1}{2} c_{2}(T)\left(\phi-\phi_{c}\right)^{2}+\cdots\) Answer the following questions. (a) Show that \(c_{2}(T)\) becomes zero at \(T=T_{c}\) and therefore \(c_{2}(T)\) is written as $$ c_{2}(T)=a\left(T-T_{c}\right) $$ where \(a\) is a constant independent of \(T\). (b) If \(\phi-\phi_{c}\) is small, the free energy of the system is written as $$ A=\int d r\left[\frac{1}{2} a\left(T-T_{c}\right)\left(\phi-\phi_{c}\right)^{2}+\frac{1}{2} \kappa_{s}(\nabla \phi)^{2}\right] $$ where the gravity term is ignored. Explain why the term linear in \(\phi-\phi_{c}\) can be ignored in eq. \((8.140)\). (c) Consider that a hypothetical external field \(h(r)\) is applied to the solute segment. Show that \(\delta \phi(r)=\phi(r)-\phi_{c}\) satisfies the following equation \(\frac{\partial \delta \phi}{\partial t}=\frac{\phi_{c}}{\xi} \nabla^{2}\left[a\left(T-T_{c}\right) \delta \phi+\phi_{c} \kappa_{s} \nabla^{2} \delta \phi-h(r)\right]\) \((8.141)\) (d) Solve eq. (8.141) using the Fourier transform, and show that the dynamic scattering function \(S_{d}(\boldsymbol{k}, t)\) is given by $$ S_{d}(k, t)=\frac{k_{B} T}{a\left(T-T_{c}\right)+\phi_{c} \kappa_{s} k^{2}} \exp \left[-\alpha_{k} t\right] $$ where \(\alpha_{k}\) is given by $$ \alpha_{k}=\frac{\phi_{c} \boldsymbol{k}^{2}}{\xi}\left[a\left(T-T_{c}\right)+\phi_{c} \kappa_{s} \boldsymbol{k}^{2}\right] $$ (e) Confirm that the static scattering function \(S(k)=S_{d}(k, 0)\) is written in the form of eq. (8.21), and show that the parameters \(N_{c}\) and \(\ell_{c}\) diverge as $$ N_{c} \propto \frac{1}{T-T_{c}}, \quad \ell_{c} \propto \frac{1}{\left(T-T_{c}\right)^{1 / 2}} $$
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